Properties

Label 3024.2.q.c.2881.1
Level $3024$
Weight $2$
Character 3024.2881
Analytic conductor $24.147$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3024,2,Mod(2305,3024)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3024, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3024.2305");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3024.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(24.1467615712\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 504)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 2881.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 3024.2881
Dual form 3024.2.q.c.2305.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.500000 + 0.866025i) q^{5} +(2.00000 - 1.73205i) q^{7} +O(q^{10})\) \(q+(-0.500000 + 0.866025i) q^{5} +(2.00000 - 1.73205i) q^{7} +(-1.50000 - 2.59808i) q^{11} +(-1.50000 - 2.59808i) q^{13} +(-2.50000 + 4.33013i) q^{17} +(3.50000 + 6.06218i) q^{19} +(-2.50000 + 4.33013i) q^{23} +(2.00000 + 3.46410i) q^{25} +(-0.500000 + 0.866025i) q^{29} +8.00000 q^{31} +(0.500000 + 2.59808i) q^{35} +(-1.50000 - 2.59808i) q^{37} +(-2.50000 - 4.33013i) q^{41} +(-3.50000 + 6.06218i) q^{43} +8.00000 q^{47} +(1.00000 - 6.92820i) q^{49} +(-0.500000 + 0.866025i) q^{53} +3.00000 q^{55} +10.0000 q^{61} +3.00000 q^{65} +12.0000 q^{67} +12.0000 q^{71} +(2.50000 - 4.33013i) q^{73} +(-7.50000 - 2.59808i) q^{77} +8.00000 q^{79} +(7.50000 - 12.9904i) q^{83} +(-2.50000 - 4.33013i) q^{85} +(-2.50000 - 4.33013i) q^{89} +(-7.50000 - 2.59808i) q^{91} -7.00000 q^{95} +(-3.50000 + 6.06218i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{5} + 4 q^{7} - 3 q^{11} - 3 q^{13} - 5 q^{17} + 7 q^{19} - 5 q^{23} + 4 q^{25} - q^{29} + 16 q^{31} + q^{35} - 3 q^{37} - 5 q^{41} - 7 q^{43} + 16 q^{47} + 2 q^{49} - q^{53} + 6 q^{55} + 20 q^{61} + 6 q^{65} + 24 q^{67} + 24 q^{71} + 5 q^{73} - 15 q^{77} + 16 q^{79} + 15 q^{83} - 5 q^{85} - 5 q^{89} - 15 q^{91} - 14 q^{95} - 7 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1135\) \(2593\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{2}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −0.500000 + 0.866025i −0.223607 + 0.387298i −0.955901 0.293691i \(-0.905116\pi\)
0.732294 + 0.680989i \(0.238450\pi\)
\(6\) 0 0
\(7\) 2.00000 1.73205i 0.755929 0.654654i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.50000 2.59808i −0.452267 0.783349i 0.546259 0.837616i \(-0.316051\pi\)
−0.998526 + 0.0542666i \(0.982718\pi\)
\(12\) 0 0
\(13\) −1.50000 2.59808i −0.416025 0.720577i 0.579510 0.814965i \(-0.303244\pi\)
−0.995535 + 0.0943882i \(0.969911\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.50000 + 4.33013i −0.606339 + 1.05021i 0.385499 + 0.922708i \(0.374029\pi\)
−0.991838 + 0.127502i \(0.959304\pi\)
\(18\) 0 0
\(19\) 3.50000 + 6.06218i 0.802955 + 1.39076i 0.917663 + 0.397360i \(0.130073\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.50000 + 4.33013i −0.521286 + 0.902894i 0.478407 + 0.878138i \(0.341214\pi\)
−0.999694 + 0.0247559i \(0.992119\pi\)
\(24\) 0 0
\(25\) 2.00000 + 3.46410i 0.400000 + 0.692820i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −0.500000 + 0.866025i −0.0928477 + 0.160817i −0.908708 0.417432i \(-0.862930\pi\)
0.815861 + 0.578249i \(0.196264\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.500000 + 2.59808i 0.0845154 + 0.439155i
\(36\) 0 0
\(37\) −1.50000 2.59808i −0.246598 0.427121i 0.715981 0.698119i \(-0.245980\pi\)
−0.962580 + 0.270998i \(0.912646\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.50000 4.33013i −0.390434 0.676252i 0.602072 0.798441i \(-0.294342\pi\)
−0.992507 + 0.122189i \(0.961009\pi\)
\(42\) 0 0
\(43\) −3.50000 + 6.06218i −0.533745 + 0.924473i 0.465478 + 0.885059i \(0.345882\pi\)
−0.999223 + 0.0394140i \(0.987451\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) 1.00000 6.92820i 0.142857 0.989743i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −0.500000 + 0.866025i −0.0686803 + 0.118958i −0.898321 0.439340i \(-0.855212\pi\)
0.829640 + 0.558298i \(0.188546\pi\)
\(54\) 0 0
\(55\) 3.00000 0.404520
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.00000 0.372104
\(66\) 0 0
\(67\) 12.0000 1.46603 0.733017 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) 2.50000 4.33013i 0.292603 0.506803i −0.681822 0.731519i \(-0.738812\pi\)
0.974424 + 0.224716i \(0.0721453\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −7.50000 2.59808i −0.854704 0.296078i
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 7.50000 12.9904i 0.823232 1.42588i −0.0800311 0.996792i \(-0.525502\pi\)
0.903263 0.429087i \(-0.141165\pi\)
\(84\) 0 0
\(85\) −2.50000 4.33013i −0.271163 0.469668i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −2.50000 4.33013i −0.264999 0.458993i 0.702564 0.711621i \(-0.252038\pi\)
−0.967563 + 0.252628i \(0.918705\pi\)
\(90\) 0 0
\(91\) −7.50000 2.59808i −0.786214 0.272352i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −7.00000 −0.718185
\(96\) 0 0
\(97\) −3.50000 + 6.06218i −0.355371 + 0.615521i −0.987181 0.159602i \(-0.948979\pi\)
0.631810 + 0.775123i \(0.282312\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −4.50000 7.79423i −0.447767 0.775555i 0.550474 0.834853i \(-0.314447\pi\)
−0.998240 + 0.0592978i \(0.981114\pi\)
\(102\) 0 0
\(103\) −7.50000 + 12.9904i −0.738997 + 1.27998i 0.213950 + 0.976845i \(0.431367\pi\)
−0.952947 + 0.303136i \(0.901966\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.50000 + 7.79423i 0.435031 + 0.753497i 0.997298 0.0734594i \(-0.0234039\pi\)
−0.562267 + 0.826956i \(0.690071\pi\)
\(108\) 0 0
\(109\) −3.50000 + 6.06218i −0.335239 + 0.580651i −0.983531 0.180741i \(-0.942150\pi\)
0.648292 + 0.761392i \(0.275484\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 7.50000 + 12.9904i 0.705541 + 1.22203i 0.966496 + 0.256681i \(0.0826291\pi\)
−0.260955 + 0.965351i \(0.584038\pi\)
\(114\) 0 0
\(115\) −2.50000 4.33013i −0.233126 0.403786i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 2.50000 + 12.9904i 0.229175 + 1.19083i
\(120\) 0 0
\(121\) 1.00000 1.73205i 0.0909091 0.157459i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −9.00000 −0.804984
\(126\) 0 0
\(127\) 4.00000 0.354943 0.177471 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 3.50000 6.06218i 0.305796 0.529655i −0.671642 0.740876i \(-0.734411\pi\)
0.977438 + 0.211221i \(0.0677440\pi\)
\(132\) 0 0
\(133\) 17.5000 + 6.06218i 1.51744 + 0.525657i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 11.5000 + 19.9186i 0.982511 + 1.70176i 0.652512 + 0.757778i \(0.273715\pi\)
0.329999 + 0.943981i \(0.392952\pi\)
\(138\) 0 0
\(139\) −4.50000 7.79423i −0.381685 0.661098i 0.609618 0.792695i \(-0.291323\pi\)
−0.991303 + 0.131597i \(0.957989\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −4.50000 + 7.79423i −0.376309 + 0.651786i
\(144\) 0 0
\(145\) −0.500000 0.866025i −0.0415227 0.0719195i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −2.50000 + 4.33013i −0.204808 + 0.354738i −0.950072 0.312032i \(-0.898990\pi\)
0.745264 + 0.666770i \(0.232324\pi\)
\(150\) 0 0
\(151\) −2.50000 4.33013i −0.203447 0.352381i 0.746190 0.665733i \(-0.231881\pi\)
−0.949637 + 0.313353i \(0.898548\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −4.00000 + 6.92820i −0.321288 + 0.556487i
\(156\) 0 0
\(157\) −6.00000 −0.478852 −0.239426 0.970915i \(-0.576959\pi\)
−0.239426 + 0.970915i \(0.576959\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 2.50000 + 12.9904i 0.197028 + 1.02379i
\(162\) 0 0
\(163\) −2.50000 4.33013i −0.195815 0.339162i 0.751352 0.659901i \(-0.229402\pi\)
−0.947167 + 0.320740i \(0.896069\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 10.5000 + 18.1865i 0.812514 + 1.40732i 0.911099 + 0.412188i \(0.135235\pi\)
−0.0985846 + 0.995129i \(0.531432\pi\)
\(168\) 0 0
\(169\) 2.00000 3.46410i 0.153846 0.266469i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −2.00000 −0.152057 −0.0760286 0.997106i \(-0.524224\pi\)
−0.0760286 + 0.997106i \(0.524224\pi\)
\(174\) 0 0
\(175\) 10.0000 + 3.46410i 0.755929 + 0.261861i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −2.50000 + 4.33013i −0.186859 + 0.323649i −0.944201 0.329369i \(-0.893164\pi\)
0.757343 + 0.653018i \(0.226497\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 3.00000 0.220564
\(186\) 0 0
\(187\) 15.0000 1.09691
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 16.0000 1.15772 0.578860 0.815427i \(-0.303498\pi\)
0.578860 + 0.815427i \(0.303498\pi\)
\(192\) 0 0
\(193\) −18.0000 −1.29567 −0.647834 0.761781i \(-0.724325\pi\)
−0.647834 + 0.761781i \(0.724325\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) −9.50000 + 16.4545i −0.673437 + 1.16643i 0.303486 + 0.952836i \(0.401849\pi\)
−0.976923 + 0.213591i \(0.931484\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0.500000 + 2.59808i 0.0350931 + 0.182349i
\(204\) 0 0
\(205\) 5.00000 0.349215
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 10.5000 18.1865i 0.726300 1.25799i
\(210\) 0 0
\(211\) −6.50000 11.2583i −0.447478 0.775055i 0.550743 0.834675i \(-0.314345\pi\)
−0.998221 + 0.0596196i \(0.981011\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −3.50000 6.06218i −0.238698 0.413437i
\(216\) 0 0
\(217\) 16.0000 13.8564i 1.08615 0.940634i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 15.0000 1.00901
\(222\) 0 0
\(223\) 10.5000 18.1865i 0.703132 1.21786i −0.264229 0.964460i \(-0.585118\pi\)
0.967361 0.253401i \(-0.0815490\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −5.50000 9.52628i −0.365048 0.632281i 0.623736 0.781635i \(-0.285614\pi\)
−0.988784 + 0.149354i \(0.952281\pi\)
\(228\) 0 0
\(229\) 0.500000 0.866025i 0.0330409 0.0572286i −0.849032 0.528341i \(-0.822814\pi\)
0.882073 + 0.471113i \(0.156147\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 5.50000 + 9.52628i 0.360317 + 0.624087i 0.988013 0.154371i \(-0.0493352\pi\)
−0.627696 + 0.778459i \(0.716002\pi\)
\(234\) 0 0
\(235\) −4.00000 + 6.92820i −0.260931 + 0.451946i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 12.5000 + 21.6506i 0.808558 + 1.40046i 0.913863 + 0.406023i \(0.133085\pi\)
−0.105305 + 0.994440i \(0.533582\pi\)
\(240\) 0 0
\(241\) 2.50000 + 4.33013i 0.161039 + 0.278928i 0.935242 0.354010i \(-0.115182\pi\)
−0.774202 + 0.632938i \(0.781849\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 5.50000 + 4.33013i 0.351382 + 0.276642i
\(246\) 0 0
\(247\) 10.5000 18.1865i 0.668099 1.15718i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −20.0000 −1.26239 −0.631194 0.775625i \(-0.717435\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(252\) 0 0
\(253\) 15.0000 0.943042
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −10.5000 + 18.1865i −0.654972 + 1.13444i 0.326929 + 0.945049i \(0.393986\pi\)
−0.981901 + 0.189396i \(0.939347\pi\)
\(258\) 0 0
\(259\) −7.50000 2.59808i −0.466027 0.161437i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −13.5000 23.3827i −0.832446 1.44184i −0.896093 0.443866i \(-0.853607\pi\)
0.0636476 0.997972i \(-0.479727\pi\)
\(264\) 0 0
\(265\) −0.500000 0.866025i −0.0307148 0.0531995i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 1.50000 2.59808i 0.0914566 0.158408i −0.816668 0.577108i \(-0.804181\pi\)
0.908124 + 0.418701i \(0.137514\pi\)
\(270\) 0 0
\(271\) 11.5000 + 19.9186i 0.698575 + 1.20997i 0.968960 + 0.247216i \(0.0795156\pi\)
−0.270385 + 0.962752i \(0.587151\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 6.00000 10.3923i 0.361814 0.626680i
\(276\) 0 0
\(277\) 2.50000 + 4.33013i 0.150210 + 0.260172i 0.931305 0.364241i \(-0.118672\pi\)
−0.781094 + 0.624413i \(0.785338\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 13.5000 23.3827i 0.805342 1.39489i −0.110717 0.993852i \(-0.535315\pi\)
0.916060 0.401042i \(-0.131352\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −12.5000 4.33013i −0.737852 0.255599i
\(288\) 0 0
\(289\) −4.00000 6.92820i −0.235294 0.407541i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 13.5000 + 23.3827i 0.788678 + 1.36603i 0.926777 + 0.375613i \(0.122568\pi\)
−0.138098 + 0.990419i \(0.544099\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 15.0000 0.867472
\(300\) 0 0
\(301\) 3.50000 + 18.1865i 0.201737 + 1.04825i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −5.00000 + 8.66025i −0.286299 + 0.495885i
\(306\) 0 0
\(307\) −4.00000 −0.228292 −0.114146 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −34.0000 −1.92179 −0.960897 0.276907i \(-0.910691\pi\)
−0.960897 + 0.276907i \(0.910691\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 30.0000 1.68497 0.842484 0.538721i \(-0.181092\pi\)
0.842484 + 0.538721i \(0.181092\pi\)
\(318\) 0 0
\(319\) 3.00000 0.167968
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −35.0000 −1.94745
\(324\) 0 0
\(325\) 6.00000 10.3923i 0.332820 0.576461i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 16.0000 13.8564i 0.882109 0.763928i
\(330\) 0 0
\(331\) −8.00000 −0.439720 −0.219860 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −6.00000 + 10.3923i −0.327815 + 0.567792i
\(336\) 0 0
\(337\) 6.50000 + 11.2583i 0.354078 + 0.613280i 0.986960 0.160968i \(-0.0514616\pi\)
−0.632882 + 0.774248i \(0.718128\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −12.0000 20.7846i −0.649836 1.12555i
\(342\) 0 0
\(343\) −10.0000 15.5885i −0.539949 0.841698i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 0 0
\(349\) −5.50000 + 9.52628i −0.294408 + 0.509930i −0.974847 0.222875i \(-0.928456\pi\)
0.680439 + 0.732805i \(0.261789\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −10.5000 18.1865i −0.558859 0.967972i −0.997592 0.0693543i \(-0.977906\pi\)
0.438733 0.898617i \(-0.355427\pi\)
\(354\) 0 0
\(355\) −6.00000 + 10.3923i −0.318447 + 0.551566i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −1.50000 2.59808i −0.0791670 0.137121i 0.823724 0.566991i \(-0.191893\pi\)
−0.902891 + 0.429870i \(0.858559\pi\)
\(360\) 0 0
\(361\) −15.0000 + 25.9808i −0.789474 + 1.36741i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 2.50000 + 4.33013i 0.130856 + 0.226649i
\(366\) 0 0
\(367\) −6.50000 11.2583i −0.339297 0.587680i 0.645003 0.764180i \(-0.276856\pi\)
−0.984301 + 0.176500i \(0.943523\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0.500000 + 2.59808i 0.0259587 + 0.134885i
\(372\) 0 0
\(373\) 0.500000 0.866025i 0.0258890 0.0448411i −0.852791 0.522253i \(-0.825092\pi\)
0.878680 + 0.477412i \(0.158425\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 3.00000 0.154508
\(378\) 0 0
\(379\) −36.0000 −1.84920 −0.924598 0.380945i \(-0.875599\pi\)
−0.924598 + 0.380945i \(0.875599\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 1.50000 2.59808i 0.0766464 0.132755i −0.825155 0.564907i \(-0.808912\pi\)
0.901801 + 0.432151i \(0.142245\pi\)
\(384\) 0 0
\(385\) 6.00000 5.19615i 0.305788 0.264820i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −8.50000 14.7224i −0.430967 0.746457i 0.565990 0.824412i \(-0.308494\pi\)
−0.996957 + 0.0779554i \(0.975161\pi\)
\(390\) 0 0
\(391\) −12.5000 21.6506i −0.632152 1.09492i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −4.00000 + 6.92820i −0.201262 + 0.348596i
\(396\) 0 0
\(397\) 8.50000 + 14.7224i 0.426603 + 0.738898i 0.996569 0.0827707i \(-0.0263769\pi\)
−0.569966 + 0.821668i \(0.693044\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1.50000 2.59808i 0.0749064 0.129742i −0.826139 0.563466i \(-0.809468\pi\)
0.901046 + 0.433724i \(0.142801\pi\)
\(402\) 0 0
\(403\) −12.0000 20.7846i −0.597763 1.03536i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −4.50000 + 7.79423i −0.223057 + 0.386346i
\(408\) 0 0
\(409\) 6.00000 0.296681 0.148340 0.988936i \(-0.452607\pi\)
0.148340 + 0.988936i \(0.452607\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 7.50000 + 12.9904i 0.368161 + 0.637673i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −11.5000 19.9186i −0.561812 0.973087i −0.997338 0.0729107i \(-0.976771\pi\)
0.435527 0.900176i \(-0.356562\pi\)
\(420\) 0 0
\(421\) −3.50000 + 6.06218i −0.170580 + 0.295452i −0.938623 0.344946i \(-0.887897\pi\)
0.768043 + 0.640398i \(0.221231\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −20.0000 −0.970143
\(426\) 0 0
\(427\) 20.0000 17.3205i 0.967868 0.838198i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 7.50000 12.9904i 0.361262 0.625725i −0.626907 0.779094i \(-0.715679\pi\)
0.988169 + 0.153370i \(0.0490126\pi\)
\(432\) 0 0
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −35.0000 −1.67428
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 20.0000 0.950229 0.475114 0.879924i \(-0.342407\pi\)
0.475114 + 0.879924i \(0.342407\pi\)
\(444\) 0 0
\(445\) 5.00000 0.237023
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −22.0000 −1.03824 −0.519122 0.854700i \(-0.673741\pi\)
−0.519122 + 0.854700i \(0.673741\pi\)
\(450\) 0 0
\(451\) −7.50000 + 12.9904i −0.353161 + 0.611693i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 6.00000 5.19615i 0.281284 0.243599i
\(456\) 0 0
\(457\) 22.0000 1.02912 0.514558 0.857455i \(-0.327956\pi\)
0.514558 + 0.857455i \(0.327956\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −10.5000 + 18.1865i −0.489034 + 0.847031i −0.999920 0.0126168i \(-0.995984\pi\)
0.510887 + 0.859648i \(0.329317\pi\)
\(462\) 0 0
\(463\) 17.5000 + 30.3109i 0.813294 + 1.40867i 0.910546 + 0.413407i \(0.135661\pi\)
−0.0972525 + 0.995260i \(0.531005\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −13.5000 23.3827i −0.624705 1.08202i −0.988598 0.150581i \(-0.951886\pi\)
0.363892 0.931441i \(-0.381448\pi\)
\(468\) 0 0
\(469\) 24.0000 20.7846i 1.10822 0.959744i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 21.0000 0.965581
\(474\) 0 0
\(475\) −14.0000 + 24.2487i −0.642364 + 1.11261i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −3.50000 6.06218i −0.159919 0.276988i 0.774920 0.632059i \(-0.217790\pi\)
−0.934839 + 0.355071i \(0.884457\pi\)
\(480\) 0 0
\(481\) −4.50000 + 7.79423i −0.205182 + 0.355386i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −3.50000 6.06218i −0.158927 0.275269i
\(486\) 0 0
\(487\) 18.5000 32.0429i 0.838315 1.45200i −0.0529875 0.998595i \(-0.516874\pi\)
0.891303 0.453409i \(-0.149792\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 10.5000 + 18.1865i 0.473858 + 0.820747i 0.999552 0.0299272i \(-0.00952753\pi\)
−0.525694 + 0.850674i \(0.676194\pi\)
\(492\) 0 0
\(493\) −2.50000 4.33013i −0.112594 0.195019i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 24.0000 20.7846i 1.07655 0.932317i
\(498\) 0 0
\(499\) 0.500000 0.866025i 0.0223831 0.0387686i −0.854617 0.519259i \(-0.826208\pi\)
0.877000 + 0.480490i \(0.159541\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 40.0000 1.78351 0.891756 0.452517i \(-0.149474\pi\)
0.891756 + 0.452517i \(0.149474\pi\)
\(504\) 0 0
\(505\) 9.00000 0.400495
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −10.5000 + 18.1865i −0.465404 + 0.806104i −0.999220 0.0394971i \(-0.987424\pi\)
0.533815 + 0.845601i \(0.320758\pi\)
\(510\) 0 0
\(511\) −2.50000 12.9904i −0.110593 0.574661i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −7.50000 12.9904i −0.330489 0.572425i
\(516\) 0 0
\(517\) −12.0000 20.7846i −0.527759 0.914106i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 1.50000 2.59808i 0.0657162 0.113824i −0.831295 0.555831i \(-0.812400\pi\)
0.897011 + 0.442007i \(0.145733\pi\)
\(522\) 0 0
\(523\) −12.5000 21.6506i −0.546587 0.946716i −0.998505 0.0546569i \(-0.982594\pi\)
0.451918 0.892059i \(-0.350740\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −20.0000 + 34.6410i −0.871214 + 1.50899i
\(528\) 0 0
\(529\) −1.00000 1.73205i −0.0434783 0.0753066i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −7.50000 + 12.9904i −0.324861 + 0.562676i
\(534\) 0 0
\(535\) −9.00000 −0.389104
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −19.5000 + 7.79423i −0.839924 + 0.335721i
\(540\) 0 0
\(541\) −15.5000 26.8468i −0.666397 1.15423i −0.978905 0.204318i \(-0.934502\pi\)
0.312507 0.949915i \(-0.398831\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −3.50000 6.06218i −0.149924 0.259675i
\(546\) 0 0
\(547\) −17.5000 + 30.3109i −0.748246 + 1.29600i 0.200417 + 0.979711i \(0.435770\pi\)
−0.948663 + 0.316289i \(0.897563\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −7.00000 −0.298210
\(552\) 0 0
\(553\) 16.0000 13.8564i 0.680389 0.589234i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 9.50000 16.4545i 0.402528 0.697199i −0.591502 0.806303i \(-0.701465\pi\)
0.994030 + 0.109104i \(0.0347983\pi\)
\(558\) 0 0
\(559\) 21.0000 0.888205
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 4.00000 0.168580 0.0842900 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) 0 0
\(565\) −15.0000 −0.631055
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 0 0
\(571\) −32.0000 −1.33916 −0.669579 0.742741i \(-0.733526\pi\)
−0.669579 + 0.742741i \(0.733526\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −20.0000 −0.834058
\(576\) 0 0
\(577\) 4.50000 7.79423i 0.187337 0.324478i −0.757024 0.653387i \(-0.773348\pi\)
0.944362 + 0.328909i \(0.106681\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −7.50000 38.9711i −0.311152 1.61680i
\(582\) 0 0
\(583\) 3.00000 0.124247
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 17.5000 30.3109i 0.722302 1.25106i −0.237773 0.971321i \(-0.576417\pi\)
0.960075 0.279743i \(-0.0902494\pi\)
\(588\) 0 0
\(589\) 28.0000 + 48.4974i 1.15372 + 1.99830i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −16.5000 28.5788i −0.677574 1.17359i −0.975709 0.219069i \(-0.929698\pi\)
0.298136 0.954524i \(-0.403635\pi\)
\(594\) 0 0
\(595\) −12.5000 4.33013i −0.512450 0.177518i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 16.0000 0.653742 0.326871 0.945069i \(-0.394006\pi\)
0.326871 + 0.945069i \(0.394006\pi\)
\(600\) 0 0
\(601\) −3.50000 + 6.06218i −0.142768 + 0.247281i −0.928538 0.371237i \(-0.878934\pi\)
0.785770 + 0.618519i \(0.212267\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.00000 + 1.73205i 0.0406558 + 0.0704179i
\(606\) 0 0
\(607\) −11.5000 + 19.9186i −0.466771 + 0.808470i −0.999279 0.0379540i \(-0.987916\pi\)
0.532509 + 0.846424i \(0.321249\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −12.0000 20.7846i −0.485468 0.840855i
\(612\) 0 0
\(613\) −21.5000 + 37.2391i −0.868377 + 1.50407i −0.00472215 + 0.999989i \(0.501503\pi\)
−0.863655 + 0.504084i \(0.831830\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 13.5000 + 23.3827i 0.543490 + 0.941351i 0.998700 + 0.0509678i \(0.0162306\pi\)
−0.455211 + 0.890384i \(0.650436\pi\)
\(618\) 0 0
\(619\) 7.50000 + 12.9904i 0.301450 + 0.522127i 0.976465 0.215677i \(-0.0691959\pi\)
−0.675014 + 0.737805i \(0.735863\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −12.5000 4.33013i −0.500802 0.173483i
\(624\) 0 0
\(625\) −5.50000 + 9.52628i −0.220000 + 0.381051i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 15.0000 0.598089
\(630\) 0 0
\(631\) 40.0000 1.59237 0.796187 0.605050i \(-0.206847\pi\)
0.796187 + 0.605050i \(0.206847\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −2.00000 + 3.46410i −0.0793676 + 0.137469i
\(636\) 0 0
\(637\) −19.5000 + 7.79423i −0.772618 + 0.308819i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −8.50000 14.7224i −0.335730 0.581501i 0.647895 0.761730i \(-0.275650\pi\)
−0.983625 + 0.180229i \(0.942316\pi\)
\(642\) 0 0
\(643\) 21.5000 + 37.2391i 0.847877 + 1.46857i 0.883099 + 0.469187i \(0.155453\pi\)
−0.0352216 + 0.999380i \(0.511214\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −4.50000 + 7.79423i −0.176913 + 0.306423i −0.940822 0.338902i \(-0.889945\pi\)
0.763908 + 0.645325i \(0.223278\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −6.50000 + 11.2583i −0.254365 + 0.440573i −0.964723 0.263268i \(-0.915200\pi\)
0.710358 + 0.703840i \(0.248533\pi\)
\(654\) 0 0
\(655\) 3.50000 + 6.06218i 0.136756 + 0.236869i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −14.5000 + 25.1147i −0.564840 + 0.978331i 0.432225 + 0.901766i \(0.357729\pi\)
−0.997065 + 0.0765653i \(0.975605\pi\)
\(660\) 0 0
\(661\) −46.0000 −1.78919 −0.894596 0.446875i \(-0.852537\pi\)
−0.894596 + 0.446875i \(0.852537\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −14.0000 + 12.1244i −0.542897 + 0.470162i
\(666\) 0 0
\(667\) −2.50000 4.33013i −0.0968004 0.167663i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −15.0000 25.9808i −0.579069 1.00298i
\(672\) 0 0
\(673\) 14.5000 25.1147i 0.558934 0.968102i −0.438652 0.898657i \(-0.644544\pi\)
0.997586 0.0694449i \(-0.0221228\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −18.0000 −0.691796 −0.345898 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) 0 0
\(679\) 3.50000 + 18.1865i 0.134318 + 0.697935i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 19.5000 33.7750i 0.746147 1.29236i −0.203510 0.979073i \(-0.565235\pi\)
0.949657 0.313291i \(-0.101432\pi\)
\(684\) 0 0
\(685\) −23.0000 −0.878785
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 3.00000 0.114291
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 9.00000 0.341389
\(696\) 0 0
\(697\) 25.0000 0.946943
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 2.00000 0.0755390 0.0377695 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) 0 0
\(703\) 10.5000 18.1865i 0.396015 0.685918i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −22.5000 7.79423i −0.846200 0.293132i
\(708\) 0 0
\(709\) −38.0000 −1.42712 −0.713560 0.700594i \(-0.752918\pi\)
−0.713560 + 0.700594i \(0.752918\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −20.0000 + 34.6410i −0.749006 + 1.29732i
\(714\) 0 0
\(715\) −4.50000 7.79423i −0.168290 0.291488i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 10.5000 + 18.1865i 0.391584 + 0.678243i 0.992659 0.120950i \(-0.0385939\pi\)
−0.601075 + 0.799193i \(0.705261\pi\)
\(720\) 0 0
\(721\) 7.50000 + 38.9711i 0.279315 + 1.45136i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −4.00000 −0.148556
\(726\) 0 0
\(727\) −3.50000 + 6.06218i −0.129808 + 0.224834i −0.923602 0.383353i \(-0.874769\pi\)
0.793794 + 0.608186i \(0.208103\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −17.5000 30.3109i −0.647261 1.12109i
\(732\) 0 0
\(733\) 14.5000 25.1147i 0.535570 0.927634i −0.463566 0.886062i \(-0.653430\pi\)
0.999136 0.0415715i \(-0.0132364\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −18.0000 31.1769i −0.663039 1.14842i
\(738\) 0 0
\(739\) 0.500000 0.866025i 0.0183928 0.0318573i −0.856683 0.515844i \(-0.827478\pi\)
0.875075 + 0.483987i \(0.160812\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 24.5000 + 42.4352i 0.898818 + 1.55680i 0.829007 + 0.559238i \(0.188906\pi\)
0.0698106 + 0.997560i \(0.477761\pi\)
\(744\) 0 0
\(745\) −2.50000 4.33013i −0.0915929 0.158644i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 22.5000 + 7.79423i 0.822132 + 0.284795i
\(750\) 0 0
\(751\) −3.50000 + 6.06218i −0.127717 + 0.221212i −0.922792 0.385299i \(-0.874098\pi\)
0.795075 + 0.606511i \(0.207432\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 5.00000 0.181969
\(756\) 0 0
\(757\) −38.0000 −1.38113 −0.690567 0.723269i \(-0.742639\pi\)
−0.690567 + 0.723269i \(0.742639\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 13.5000 23.3827i 0.489375 0.847622i −0.510551 0.859848i \(-0.670558\pi\)
0.999925 + 0.0122260i \(0.00389175\pi\)
\(762\) 0 0
\(763\) 3.50000 + 18.1865i 0.126709 + 0.658397i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −15.5000 26.8468i −0.558944 0.968120i −0.997585 0.0694574i \(-0.977873\pi\)
0.438641 0.898663i \(-0.355460\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −8.50000 + 14.7224i −0.305724 + 0.529529i −0.977422 0.211296i \(-0.932232\pi\)
0.671698 + 0.740825i \(0.265565\pi\)
\(774\) 0 0
\(775\) 16.0000 + 27.7128i 0.574737 + 0.995474i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 17.5000 30.3109i 0.627003 1.08600i
\(780\) 0 0
\(781\) −18.0000 31.1769i −0.644091 1.11560i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 3.00000 5.19615i 0.107075 0.185459i
\(786\) 0 0
\(787\) −52.0000 −1.85360 −0.926800 0.375555i \(-0.877452\pi\)
−0.926800 + 0.375555i \(0.877452\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 37.5000 + 12.9904i 1.33335 + 0.461885i
\(792\) 0 0
\(793\) −15.0000 25.9808i −0.532666 0.922604i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −16.5000 28.5788i −0.584460 1.01231i −0.994943 0.100446i \(-0.967973\pi\)
0.410483 0.911868i \(-0.365360\pi\)
\(798\) 0 0
\(799\) −20.0000 + 34.6410i −0.707549 + 1.22551i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −15.0000 −0.529339
\(804\) 0 0
\(805\) −12.5000 4.33013i −0.440567 0.152617i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −16.5000 + 28.5788i −0.580109 + 1.00478i 0.415357 + 0.909659i \(0.363657\pi\)
−0.995466 + 0.0951198i \(0.969677\pi\)
\(810\) 0 0
\(811\) −12.0000 −0.421377 −0.210688 0.977553i \(-0.567571\pi\)
−0.210688 + 0.977553i \(0.567571\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 5.00000 0.175142
\(816\) 0 0
\(817\) −49.0000 −1.71429
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 42.0000 1.46581 0.732905 0.680331i \(-0.238164\pi\)
0.732905 + 0.680331i \(0.238164\pi\)
\(822\) 0 0
\(823\) −8.00000 −0.278862 −0.139431 0.990232i \(-0.544527\pi\)
−0.139431 + 0.990232i \(0.544527\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 36.0000 1.25184 0.625921 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) 0 0
\(829\) 12.5000 21.6506i 0.434143 0.751958i −0.563082 0.826401i \(-0.690385\pi\)
0.997225 + 0.0744432i \(0.0237179\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 27.5000 + 21.6506i 0.952819 + 0.750150i
\(834\) 0 0
\(835\) −21.0000 −0.726735
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1.50000 2.59808i 0.0517858 0.0896956i −0.838971 0.544177i \(-0.816842\pi\)
0.890756 + 0.454481i \(0.150175\pi\)
\(840\) 0 0
\(841\) 14.0000 + 24.2487i 0.482759 + 0.836162i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 2.00000 + 3.46410i 0.0688021 + 0.119169i
\(846\) 0 0
\(847\) −1.00000 5.19615i −0.0343604 0.178542i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 15.0000 0.514193
\(852\) 0 0
\(853\) 26.5000 45.8993i 0.907343 1.57156i 0.0896015 0.995978i \(-0.471441\pi\)
0.817741 0.575586i \(-0.195226\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −22.5000 38.9711i −0.768585 1.33123i −0.938330 0.345741i \(-0.887628\pi\)
0.169745 0.985488i \(-0.445706\pi\)
\(858\) 0 0
\(859\) −11.5000 + 19.9186i −0.392375 + 0.679613i −0.992762 0.120096i \(-0.961680\pi\)
0.600387 + 0.799709i \(0.295013\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −23.5000 40.7032i −0.799949 1.38555i −0.919648 0.392743i \(-0.871526\pi\)
0.119699 0.992810i \(-0.461807\pi\)
\(864\) 0 0
\(865\) 1.00000 1.73205i 0.0340010 0.0588915i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −12.0000 20.7846i −0.407072 0.705070i
\(870\) 0 0
\(871\) −18.0000 31.1769i −0.609907 1.05639i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −18.0000 + 15.5885i −0.608511 + 0.526986i
\(876\) 0 0
\(877\) 18.5000 32.0429i 0.624701 1.08201i −0.363898 0.931439i \(-0.618554\pi\)
0.988599 0.150574i \(-0.0481123\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −22.0000 −0.741199 −0.370599 0.928793i \(-0.620848\pi\)
−0.370599 + 0.928793i \(0.620848\pi\)
\(882\) 0 0
\(883\) 44.0000 1.48072 0.740359 0.672212i \(-0.234656\pi\)
0.740359 + 0.672212i \(0.234656\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1.50000 2.59808i 0.0503651 0.0872349i −0.839744 0.542983i \(-0.817295\pi\)
0.890109 + 0.455748i \(0.150628\pi\)
\(888\) 0 0
\(889\) 8.00000 6.92820i 0.268311 0.232364i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 28.0000 + 48.4974i 0.936984 + 1.62290i
\(894\) 0 0
\(895\) −2.50000 4.33013i −0.0835658 0.144740i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −4.00000 + 6.92820i −0.133407 + 0.231069i
\(900\) 0 0
\(901\) −2.50000 4.33013i −0.0832871 0.144257i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −1.00000 + 1.73205i −0.0332411 + 0.0575753i
\(906\) 0 0
\(907\) −18.5000 32.0429i −0.614282 1.06397i −0.990510 0.137441i \(-0.956112\pi\)
0.376228 0.926527i \(-0.377221\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −6.50000 + 11.2583i −0.215355 + 0.373005i −0.953382 0.301765i \(-0.902424\pi\)
0.738028 + 0.674771i \(0.235757\pi\)
\(912\) 0 0
\(913\) −45.0000 −1.48928
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −3.50000 18.1865i −0.115580 0.600572i
\(918\) 0 0
\(919\) −20.5000 35.5070i −0.676233 1.17127i −0.976107 0.217291i \(-0.930278\pi\)
0.299874 0.953979i \(-0.403055\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −18.0000 31.1769i −0.592477 1.02620i
\(924\) 0 0
\(925\) 6.00000 10.3923i 0.197279 0.341697i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −34.0000 −1.11550 −0.557752 0.830008i \(-0.688336\pi\)
−0.557752 + 0.830008i \(0.688336\pi\)
\(930\) 0 0
\(931\) 45.5000 18.1865i 1.49120 0.596040i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −7.50000 + 12.9904i −0.245276 + 0.424831i
\(936\) 0 0
\(937\) −6.00000 −0.196011 −0.0980057 0.995186i \(-0.531246\pi\)
−0.0980057 + 0.995186i \(0.531246\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 30.0000 0.977972 0.488986 0.872292i \(-0.337367\pi\)
0.488986 + 0.872292i \(0.337367\pi\)
\(942\) 0 0
\(943\) 25.0000 0.814112
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 52.0000 1.68977 0.844886 0.534946i \(-0.179668\pi\)
0.844886 + 0.534946i \(0.179668\pi\)
\(948\) 0 0
\(949\) −15.0000 −0.486921
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −46.0000 −1.49009 −0.745043 0.667016i \(-0.767571\pi\)
−0.745043 + 0.667016i \(0.767571\pi\)
\(954\) 0 0
\(955\) −8.00000 + 13.8564i −0.258874 + 0.448383i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 57.5000 + 19.9186i 1.85677 + 0.643205i
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 9.00000 15.5885i 0.289720 0.501810i
\(966\) 0 0
\(967\) −10.5000 18.1865i −0.337657 0.584839i 0.646334 0.763054i \(-0.276301\pi\)
−0.983992 + 0.178215i \(0.942968\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −3.50000 6.06218i −0.112320 0.194545i 0.804385 0.594108i \(-0.202495\pi\)
−0.916705 + 0.399564i \(0.869162\pi\)
\(972\) 0 0
\(973\) −22.5000 7.79423i −0.721317 0.249871i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −18.0000 −0.575871 −0.287936 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) 0 0
\(979\) −7.50000 + 12.9904i −0.239701 + 0.415174i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −13.5000 23.3827i −0.430583 0.745792i 0.566340 0.824171i \(-0.308359\pi\)
−0.996924 + 0.0783795i \(0.975025\pi\)
\(984\) 0 0
\(985\) −3.00000 + 5.19615i −0.0955879 + 0.165563i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −17.5000 30.3109i −0.556468 0.963830i
\(990\) 0 0
\(991\) 10.5000 18.1865i 0.333543 0.577714i −0.649660 0.760224i \(-0.725089\pi\)
0.983204 + 0.182510i \(0.0584223\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −9.50000 16.4545i −0.301170 0.521642i
\(996\) 0 0
\(997\) 8.50000 + 14.7224i 0.269198 + 0.466264i 0.968655 0.248410i \(-0.0799082\pi\)
−0.699457 + 0.714675i \(0.746575\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 3024.2.q.c.2881.1 2
3.2 odd 2 1008.2.q.d.529.1 2
4.3 odd 2 1512.2.q.a.1369.1 2
7.2 even 3 3024.2.t.e.289.1 2
9.4 even 3 3024.2.t.e.1873.1 2
9.5 odd 6 1008.2.t.a.193.1 2
12.11 even 2 504.2.q.b.25.1 2
21.2 odd 6 1008.2.t.a.961.1 2
28.23 odd 6 1512.2.t.b.289.1 2
36.23 even 6 504.2.t.b.193.1 yes 2
36.31 odd 6 1512.2.t.b.361.1 2
63.23 odd 6 1008.2.q.d.625.1 2
63.58 even 3 inner 3024.2.q.c.2305.1 2
84.23 even 6 504.2.t.b.457.1 yes 2
252.23 even 6 504.2.q.b.121.1 yes 2
252.247 odd 6 1512.2.q.a.793.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.q.b.25.1 2 12.11 even 2
504.2.q.b.121.1 yes 2 252.23 even 6
504.2.t.b.193.1 yes 2 36.23 even 6
504.2.t.b.457.1 yes 2 84.23 even 6
1008.2.q.d.529.1 2 3.2 odd 2
1008.2.q.d.625.1 2 63.23 odd 6
1008.2.t.a.193.1 2 9.5 odd 6
1008.2.t.a.961.1 2 21.2 odd 6
1512.2.q.a.793.1 2 252.247 odd 6
1512.2.q.a.1369.1 2 4.3 odd 2
1512.2.t.b.289.1 2 28.23 odd 6
1512.2.t.b.361.1 2 36.31 odd 6
3024.2.q.c.2305.1 2 63.58 even 3 inner
3024.2.q.c.2881.1 2 1.1 even 1 trivial
3024.2.t.e.289.1 2 7.2 even 3
3024.2.t.e.1873.1 2 9.4 even 3